Regarding a potential generalisation of Krylov-Bogolubov theorem Let $(\mathbb{X},d)$ be a Compact Metric Space and $T:(\mathbb{X},d) \longrightarrow (\mathbb{X},d)$ be a Continuous Mapping. Let $B(\mathbb{X})$ be $\sigma$ - Algebra of all Borel subsets of $\mathbb{X}$.
It is well-known in Ergodic Theory that Krylov-Bogolubov theorem guarantees the existence of $T-$ invariant borel probability measure $\mu$ defined on $B(\mathbb{X})$ ($T-$ invariant means $\mu(B) = \mu(T^{-1}(B))$ for every $B\in B(\mathbb{X})$).
Via Riesz Representation Theorem we identify the set $M(\mathbb{X})$ (the set of all borel probability measures defined on $B(\mathbb{X})$) with a subset of the set of all positive normalised continuous linear functionals on $C(\mathbb{X})$ (the space of all real continuous functions defined on $\mathbb{X}$).
Since Riesz Representation Theorem has been generalised for a Locally Compact Hausdorff topological space (that is called Riesz-Markov-Kakutani theorem).
I would like to enquire whether it is possible to generalise Krylov-Bogolubov theorem for a Locally Compact Hausdorff topological space.
To understand the whole image, you can consult An Introduction To Ergodic Theory, Peter Walters, 1982. Chapter 6.
 A: First of all, you need to formulate the appropriate generalization. Here I will assume you want the theorem for any $T: X \to X$ which is continuous, or you maybe even assume it to be invertible. There is no hope to find an invariant probability measure, so you may want to just find an invariant positive (Radon) measure. This is still not possible in general:
Take $X = \mathbb{R}$ and $T(x) = 2x$. This is a homeomorphism. It has no invariant positive measure: If $T_* \mu = \mu$ is invariant, let us assume some finite open interval $(a,b)$ has finite measure (otherwise all open sets have measure infinity, contradicting the requirement from a Radon measure to be finite on compact sets). WLOG $a > 0$. We claim $\mu((0,b))=\infty$, again contradicting fineness on compact sets. Indeed, $\mu((\frac{a}{2^n}, \frac{b}{2^n}))) = \mu((a,b))$ for all $n \geq 0$, and, these intervals are contained in $(0,b)$ yet an infinite number of them are disjoint. This shows the claim. $\blacksquare$
If you want to know more about dynamics with infinite positive measrues, look up infinite ergodic theory.
